where \(\mathcal {A}^N\)  is the \(N\) -fold cartesian product of \(\mathcal {A}\) .
In [1]}, conditions on the parameters \(b,f\)  and \(g\)  are given to guarantee that, if the problem (REF ) admits a solution \(\hat{\alpha }^N=(\hat{\alpha }^{1,N},\dots , \hat{\alpha }^{N,N})\) , then for each \(i\) , and each \(t\) , the sequence \((\hat{\alpha }^{i,N}_t)\)  converges to \(\hat{\alpha }_t\)  in \(\mathbb {L}^2\) , where \(\hat{\alpha }\)  solves the McKean-Vlasov control problem
\({\left\lbrace \begin{array}{ll}\inf _{\alpha \in \mathcal {A}}E\Big [\int _0^Tf(t, X^{\alpha }_t, \alpha _t, \mathcal {L}(X^{\alpha }_t,\alpha _t))\,dt + g(X^\alpha _T, \mathcal {L}(X^\alpha _T)) \Big ]\\dX^\alpha _t = b(t, X^\alpha _t,\alpha _t,\mathcal {L}(X^\alpha _t, \alpha _t))\,dt + \sigma \,dW_t,\quad X^\alpha _0 =x.\end{array}\right.}\) 
